Let's say that I have an infected PC. I download a copy of the Windows ISO(obviously clean) and burn it to an USB. Is it possible that the malware on my computer can inject itself in this process/ISO and somehow land itself onto my bootable USB such that even if I reinstall Windows, my PC remains infected?
It is technically possible to write some malware that intercepts the ISO extraction process and then writes a boot sector to the media in order to run some malware before the normal boot sequence to hinder detection.
I do not know whether there are any current occurrences of this type of malware in the wild however. It is probably not worth worrying about unless you would be the target of such an attack because it is not a commonly used technique. Most viruses propagate because they use common attack vectors that are known to succeed on a large scale, and such an attack would be technically harder to achieve (both ISO interception and the installation of a bootloader to launch in memory persistence) so these things are unlikely outside of a targeted attack.
Is it possible that the malware on my computer can inject itself in this process/ISO?
Practically, I do not remember hearing a virus infecting an ISO file.
Theoretically speaking however, nothing can prevent a virus from infecting an ISO file because, after all, an ISO file is just an archived file. Regarding this fact, viruses have always infected this file family (e.g. A virus has been found in an archive)
... and somehow land itself onto my bootable USB ...?
Well, you said previously:
Let's say that I have an infected PC. I download a copy of the Windows ISO(obviously clean) and burn it to an USB.
Most viruses can infect your USB already by the simple fact your machine (PC) performs a read operation over the USB stick, so you are already in trouble.
Coming back to the title of your question, a virus can easily modify the inde of any archived file and inject itself anywhere such as in the AUTORUN file available in ISO files.
Also you have to consider the case where the ISO file was already infected before you download it (yes, it is common: such ISO files -mainly Windows- are available on Internet -thepiratebay, as an example-, they run normally as an operating system but you can find within them all sorts of malware)
Also in extreme but feasible cases, your ISO file may be compromised -but not infected- when you try to download it and that is why checking the checksum files of the downloaded ISO file is often offered along with GPG files for more serious authors (Kali Linux, for example)